Find the antiderivative hidden inside an inverse-sine expression

Key concept: This kind of problem is really about matching a given antiderivative form to the integrand. A substitution with $u=\sqrt{x}$ makes the square roots much less intimidating, and then the inverse trig term gives away what the derivative must look like.

Step by step

We are given

$\int \frac{1}{x}\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}\,dx = 2f(x)-2\sin^{-1}\sqrt{x}+c.$

Differentiate both sides with respect to $x$:

$\frac{1}{x}\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}=2f'(x)-2\cdot \frac{1}{\sqrt{1-(\sqrt{x})^2}}\cdot \frac{1}{2\sqrt{x}}.$

Since $(\sqrt{x})^2=x$, this becomes

$\frac{1}{x}\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}=2f'(x)-\frac{1}{\sqrt{x}\sqrt{1-x}}.$

Now simplify the left-hand side. Let $u=\sqrt{x}$, so $x=u^2$. Then

$\sqrt{\frac{1-u}{1+u}}=\frac{\sqrt{1-u^2}}{1+u}$

and because $1-u^2=1-x$, the expression is consistent with the derivative pattern above. Matching the antiderivative structure gives

$2f'(x)=\frac{1}{x}\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}+\frac{1}{\sqrt{x}\sqrt{1-x}}.$

A cleaner route is to recognize that the given identity is arranged so the inverse-sine term accounts for the standard derivative

$\frac{d}{dx}\bigl(2\sin^{-1}\sqrt{x}\bigr)=\frac{1}{\sqrt{x}\sqrt{1-x}}.$

Then the remaining part must be $2f'(x)$, which simplifies to

$f'(x)=\frac{1}{2x}\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}.$

Integrating that expression gives

$\boxed{f(x)=\frac12\sin^{-1}\sqrt{x}+C}$

up to an additive constant absorbed into $c$.

Pitfall alert

The main trap is differentiating $\sin^{-1}(\sqrt{x})$ incorrectly. You need both the outer derivative of $\sin^{-1}$ and the inner derivative of $\sqrt{x}$. Also, don’t forget that constants of integration can shift between the two sides of the identity.

Try different conditions

If the square root were $\sqrt{\frac{1+ \sqrt{x}}{1-\sqrt{x}}}$ instead, the same strategy would still work, but the matching inverse-trig term would usually change sign or become a related inverse hyperbolic form. The key is to differentiate the identity first, then isolate $f'(x)$.

Further reading

antiderivative, inverse sine, substitution